A Two Sample Test for Functional Data

  • Lee, Jong Soo (Department of Mathematical Sciences, University of Massachusetts Lowell) ;
  • Cox, Dennis D. (Department of Statistics, Rice University) ;
  • Follen, Michele (Department of Obstetrics and Gynecology, Brookdale University Hospital and Medical Center)
  • Received : 2014.11.24
  • Accepted : 2015.01.31
  • Published : 2015.03.31


We consider testing equality of mean functions from two samples of functional data. A novel test based on the adaptive Neyman methodology applied to the Hotelling's T-squared statistic is proposed. Under the enlarged null hypothesis that the distributions of the two populations are the same, randomization methods are proposed to find a null distribution which gives accurate significance levels. An extensive simulation study is presented which shows that the proposed test works very well in comparison with several other methods under a variety of alternatives and is one of the best methods for all alternatives, whereas the other methods all show weak power at some alternatives. An application to a real-world data set demonstrates the applicability of the method.


Supported by : NIH-NCI


  1. Anderson, T. W. (2003). An Introduction to Multivariate Statistical Analysis, Third Edition, Wiley, New York.
  2. Cantor, S. B., Yamal, J. M., Guillaud, M., Cox, D. D., Atkinson, E. N., Benedet, J. L., Miller, D., Ehlen, T., Matisic, J., van Niekerk, D., Bertrand, M., Milbourne, A., Rhodes, H., Malpica, A., Staerkel, G., Nader-Eftekhari, S., Adler-Storthz, K., Scheurer, M. E., Basen-Engquist, K., Shinn, E., West, L. A., Vlastos, A. T., Tao, X., Beck, J. R., MacAulay, C. and Follen, M. (2011). Accuracy of optical spectroscopy for the detection of cervical intraepithelial neoplasia: Testing a device as an adjunct to colposcopy, International Journal of Cancer, 128, 1151-1168.
  3. Cox, D. D. and Lee, J. S. (2008). Pointwise testing with functional data using the Westfall-Young randomization method, Biometrika, 95, 621-634.
  4. Fan, J. (1996). Test of significance based on wavelet thresholding and Neyman's truncation, Journal of American Statistical Association, 91, 674-688.
  5. Fan, J. and Lin, S. (1998). Test of significance when data are curves, Journal of the American Statistical Association, 93, 1007-1021.
  6. Good, P. I. (2005). Permutation, Parametric, and Bootstrap Tests of Hypotheses, Springer, New York.
  7. Hall, P. and Hosseini-Nasab, M. (2006). On properties of functional principal components analysis, Journal of the Royal Statistical Society, Series B, 68, 109-126.
  8. Inglot, T., Kallenberg, W. C. M. and Ledwina, T. (1994). Power approximations to and power comarison of smooth goodness-of-fit tests, Scandinavian Journal of Statistics, 21, 131-145.
  9. Johnstone, I. M. (2001). On the distribution of the largest eigenvalues in principal components analysis, Annals of Statistics, 29, 295-327.
  10. Lopes, M. E., Jacob, L. and Wainwright, M. J. (2012). A more powerful two-sample test in high dimensions using random projection (arXiv:1108.2401v2)
  11. Muirhead, R. J. (1982). Aspects of Multivariate Statistical Theory, Wiley, New York.
  12. Neyman, J. (1937). Smooth test for goodness of fit, Skandinavisk Aktuarietidskrift, 20, 149-199.
  13. Pikkula, B. M., Shuhatovich, O., Price, R. L., Serachitopol, D. M., Follen, M., McKinnon, N., MacAulay, C., Richards-Kortum, R., Lee, J. S., Atkinson, E. N. and Cox, D. D. (2007). In-strumentation as a source of variability in the application of fluorescence spectroscopy devices for detecting cervical neoplasia, Journal of Biomedical Optics, 12, 034014.
  14. Ramsay, J. O. and Silverman, B. W. (2005). Functional Data Analysis, 2nd ed., Springer, New York.
  15. Rencher, A. C. (2002). Methods of Multivariate Analysis, Second edition, Wiley, New York.
  16. Shen, Q. and Faraway, J. (2004). An F test for linear models with functional responses, Statistica Sinica, 14, 1239-1257.
  17. Taylor, J. E., Worsley, K. J. and Gosselin, F. (2007). Maxima of discretely sampled random fields with an application to 'bubbles', Biometrika, 94, 1-18.
  18. Wald, A. andWolfowitz, J. (1944). Statistical tests based on permutations of the observations, Annals of Mathematical Statistics, 15, 358-372.
  19. Zhang, J. (2011). Statistical inferences for linear models with functional responses, Statistica Sinica, 21, 1431-1451